PROBLEMS 1169
PROBLEMS
1. Find the general solution:
(I' 1 1 ldp 5 3
(a) dx=( (h)i x(
d( 4 -2 x\ -1 1 \
S (3 32- dx ( - 1)
d1t 3 t) x2 )
dr ' I 1c ldx 2 - 1 1
(e) = -3 x-1 (d) =- 2 lx
S7(f) =
(2 J - ) 1
2. For each of the following choices of A determine whether the zero solution of
the equation dxi dt = Ax is stable. (It is not necessary to find the general solution.)
a) 3 4 ) -1 3
c) - 1 \ (d) 7-1 0 2
1 -3 0) 3 2
1 1 -4 2 0 -1
3. Let the following differential equation he given:
d 2 0 -3 -1 x+q(t)
dt
1 4 - 2
a) Show that the zero solution of the associated homogeneous equation is stable.
(b) Find a particular solution for q(t) = (2, 1, -8). [Hlint. Try x(t) = c, a constant
vector.
(c) Find a particular solution for q(t) = et(1, 0, 1). [Hint. Try x(t) = etc.
(d) Find a particular solution for q(t) = (0, 3 cos 2t - sin 2t, - 2 cos 2t + sin 2t).
4. Let the following differential equation he given
i1x. 5 _7
2 -2 -3 x+ q(t)
(a) Verify that for the associated homogeneous equation the eigenvalues are -1
and -i. so that the zero solution is neutrally stable.
(h) Verify that for q(t) = (sin t, 0, 0) resonance occurs.
(c) Verify that for q(t) = (2 sin t, sin t, sin t), a particular solution is given by
x = (2 sin t -2cos, sin t - cos t, sin t - cos t), so that there is no
resonance.
Benark. Parts (h) and (c) show that, even when ~iw are eigenvalues, a sinusoidal
forcing function of frequency wo may, but need not necessarily lead to resonance.
5. Coupled springs. Let na mass mI he suspended by a spring. Let a second mass n2
he suspended from the first by a second spring (Figure 14-21). Then an equilibrium
will be reached under gravity. Let x1, x2 measure the downward displacements
